Simplify and Use Square Roots

Simplify Expressions with Square Roots

Remember that when a number $n$ is multiplied by itself, we write $n^2$ and read it “$n$ squared.” For example, $15^2$ reads as “$15$ squared,” and $225$ is called the square of $15$, since $15^2 = 225$.

Sometimes we need to look at the relationship between numbers and their squares in reverse. Because $225$ is the square of $15$, we can also say that $15$ is a square root of $225$. A number whose square is $m$ is called a square root of $m$.

Notice that $(-15)^2 = 225$ also, so $-15$ is also a square root of $225$. Therefore, both $15$ and $-15$ are square roots of $225$.

So every positive number has two square roots — one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, $\sqrt{m}$, denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because $0^2 = 0$, $\sqrt{0} = 0$. Notice that zero has only one square root.

Since $15$ is the positive square root of $225$, we write $\sqrt{225} = 15$. The radical sign indicates the positive root, so $\sqrt{225} = 15$. If we want the negative square root of a number, we place a negative sign in front of the radical sign. For example, $-\sqrt{225} = -15$.

Example. Simplify: (a) $\sqrt{36}$, (b) $\sqrt{196}$, (c) $-\sqrt{81}$, (d) $-\sqrt{289}$.

Can we simplify $\sqrt{-169}$? Is there a number whose square is $-169$? Any positive number squared is positive, and any negative number squared is also positive. There is no real number equal to $\sqrt{-169}$, so we say $\sqrt{-169}$ is not a real number. Contrast this with $-\sqrt{64} = -8$, where the negative sits outside the radical.

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Example. Simplify: (a) $\sqrt{25} + \sqrt{144}$, (b) $\sqrt{25 + 144}$.

Notice the different answers in parts (a) and (b)!

Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Consider the perfect squares $4$ and $9$: since $\sqrt{4} = 2$ and $\sqrt{9} = 3$, any number between $4$ and $9$ has a square root between $2$ and $3$. For example, $\sqrt{5}$ must be between $2$ and $3$. Using inequality symbols, we write:

Example. Estimate $\sqrt{60}$ between two consecutive whole numbers.

Think of the perfect square numbers closest to $60$. The perfect squares just below and above $60$ are $49 = 7^2$ and $64 = 8^2$.

Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the $\sqrt{x}$ key on your calculator; you will use this key to approximate square roots.

When you use your calculator to find the square root of a number that is not a perfect square, the answer you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is $\approx$ and it is read “approximately.”

Suppose your calculator has a $10$-digit display. You would see that $\sqrt{5} \approx 2.236067978$. If we wanted to round $\sqrt{5}$ to two decimal places, we would say $\sqrt{5} \approx 2.24$.

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to $5$, but are not exactly equal to $5$.

Example. Round $\sqrt{17}$ to two decimal places.

Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider $\sqrt{9x^2}$. Can you think of an expression whose square is $9x^2$? Since $(3x)^2 = 9x^2$, we have $\sqrt{9x^2} = 3x$.

When we use the radical sign to take the square root of a variable expression, we should specify that $x \geq 0$ to make sure we get the principal square root. However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number, and so we will not write $x \geq 0$ next to every radical.

What about square roots of higher powers of variables? Think about the Power Property of Exponents, $\left(a^m\right)^n = a^{m \cdot n}$. If we square $a^m$, the exponent will become $2m$:

This tells us how to take square roots of even powers. For example, $\sqrt{25u^8} = 5u^4$ because $\left(5u^4\right)^2 = 25u^8$, and $\sqrt{196q^{36}} = 14q^{18}$ because $\left(14q^{18}\right)^2 = 196q^{36}$.

Example. Simplify: (a) $\sqrt{x^6}$, (b) $\sqrt{y^{16}}$.

Example. Simplify $\sqrt{16n^2}$.

Example. Simplify $-\sqrt{81c^2}$.

Example. Simplify $\sqrt{36x^2y^2}$.

Example. Simplify $\sqrt{64p^{64}}$.

Example. Simplify $\sqrt{121a^6b^8}$.

Key terms

square of a number — if $n^2 = m$, then $m$ is the square of $n$. square root of a number — if $n^2 = m$, then $n$ is a square root of $m$; every positive number has two square roots, one positive and one negative. radical sign — the symbol $\sqrt{\phantom{m}}$ that denotes the positive (principal) square root. radicand — the number or expression under the radical sign. principal square root — the non-negative square root, the one the radical sign indicates. approximation — a value close to the exact square root, written with $\approx$, accurate to the number of digits displayed.

This section is adapted from Elementary Algebra 2e, 9.1 Simplify and Use Square Roots by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the two-column worked examples as aligned step tables; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.